# Certain Fourier transforms involving Whittaker function and Bessel functions

I recently meet the following two weird "Fourier transform" questions.

(I), Suppose that $$F$$ is a $$p$$-adic field (the same question can be asked over any local field, including $$\mathbb{R}$$ and $$\mathbb{C}$$) and $$\psi$$ be a fixed nontrivial additive character of $$F$$. Let $$W$$ be a function on $$GL_2(F)$$ which satisfies the following 2 conditions:

(1), $$W\left(\begin{pmatrix} 1& x \\ &1 \end{pmatrix} g\right)=\psi(x)W(g)$$, for all $$x\in F, g\in GL_2(F)$$, and

(2), there exists an open subgroup $$K$$ of $$GL_2(F)$$ such that $$W(gk)=W(g)$$ for all $$g\in GL_2(F), k\in K$$.

One can think $$W$$ as a Whittaker function of a smooth representation of $$GL_2(F)$$. One extend the definition of $$W$$ to all $$2\times 2$$ matrices by zero extension, i.e., if $$g\in \mathrm{Mat}_{2\times 2}(F)$$ and $$\det(g)=0$$, then define $$W(g)=0$$.

Let $$\mathcal{S}(F^2)$$ be the space of Bruhat-Schwartz functions on $$F^2$$. Fix a nonzero Whittaker function $$W$$ as above. For $$\phi\in \mathcal{S}(F^2)$$, define $$\widehat{\phi}(x_1,x_2)=\int_{F^2}W\left( I_2+\begin{pmatrix} x_1\\ x_2 \end{pmatrix}\begin{pmatrix} y_1 & y_2\end{pmatrix}\right)\phi(y_1,y_2)dy_1 dy_2.$$ Here $$I_2$$ is the $$2\times 2$$ identity matrix and $$W$$ is omitted from the notation $$\widehat{\phi}$$.

Question (I): Do we know that $$\widehat \phi\in \mathcal{S}(F^2)$$ and $$\phi\mapsto \widehat\phi$$ defines an isomorphism?

This looks like a Fourier transform. Also, it is easy to see that the function $$x_1\mapsto \widehat{\phi}(x_1,0)$$ has compact support. But I don't know how to prove the above statement in general.

(II) The second question involves a Fourier transform in a similar flavor, but not the same with the above one. Suppose that $$k$$ is a finite field and $$\psi$$ is a nontrivial additive character of $$k$$. Let $$\mathscr{B}$$ be the set of functions $$B$$ on $$GL_2(k)$$ such that $$B\left(\begin{pmatrix}1&x\\ &1 \end{pmatrix} g \begin{pmatrix}1&y\\ &1 \end{pmatrix}\right)=\psi(x)\psi(-y)B(g), \forall x,y\in k, g\in GL_2(k).$$ One can think such a function $$B$$ as a Bessel function. For $$B\in \mathscr{B}$$, define \begin{align*}&\mathcal{F}_B\left(\begin{pmatrix} x_{11} & x_{12}\\ x_{21}& x_{22} \end{pmatrix}\right)\\ &\qquad :=\sum_{s_1,s_2,r_1,r_2\in k} \psi(x_{22}r_1+x_{21}r_2+s_1)B\left(\begin{pmatrix} x_{11} & x_{12}\\ x_{21}& x_{22} \end{pmatrix} \left(I_2+\begin{pmatrix} s_2\\ -s_1 \end{pmatrix}\begin{pmatrix} -r_1 & r_2\end{pmatrix} \right) \right), \end{align*} where if $$\det(g)=0$$, we view $$B(g)=0.$$

It is not hard to show that $$\mathcal{F}_B\in \mathscr{B}$$. The negative signs in the definition of $$\mathcal{F}_B$$ is to make sure $$\mathcal{F}_B\in \mathscr{B}.$$

Question (II): Is it true that $$B\mapsto \mathcal{F}_B$$ is a bijection?

If the answer of (II) is affirmative, I am also wondering if there is a local field analogue.

Question (III): Are there generalizations to $$GL_n$$?